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प्रश्न
If \[x - \frac{1}{x} = 3,\] find the values of \[x^2 + \frac{1}{x^2}\] and \[x^4 + \frac{1}{x^4} .\]
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उत्तर
Let us consider the following equation: \[x - \frac{1}{x} = 3\]
Squaring both sides, we get:
\[\left( x - \frac{1}{x} \right)^2 = \left( 3 \right)^2 = 9\]
\[ \Rightarrow \left( x - \frac{1}{x} \right)^2 = 9\]
\[ \Rightarrow x^2 - 2 \times x \times \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 9\]
\[ \Rightarrow x^2 - 2 + \frac{1}{x^2} = 9\]
\[\Rightarrow x^2 + \frac{1}{x^2} = 11\] (Adding 2 to both sides)
Squaring both sides again, we get:
\[\left( x^2 + \frac{1}{x^2} \right)^2 = \left( 11 \right)^2 = 121\]
\[ \Rightarrow \left( x^2 + \frac{1}{x^2} \right)^2 = 121\]
\[ \Rightarrow \left( x^2 \right)^2 + 2\left( x^2 \right)\left( \frac{1}{x^2} \right) + \left( \frac{1}{x^2} \right)^2 = 121\]
\[ \Rightarrow x^4 + 2 + \frac{1}{x^4} = 121\]
\[\Rightarrow x^4 + \frac{1}{x^4} = 119\]
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